The transport of scalars in turbulent flows is challenging to understand both because of the complexity of the underlying turbulent flow that advects, stretches and compresses the scalar field, and also because of the molecular diffusion to which it is subject that leads to non-trivial differences compared to the transport of fluid particles [1]}, [2]}. Indeed, while it has often been assumed that the statistical properties of passive scalars in turbulent flows should reflect the analogous properties of the underlying velocity field, e.g. a similarity in the statistical distribution of the turbulent kinetic energy and scalar dissipation rates, this is not in general the case. For example, it has been found that Kolmogorov's hypothesis of local isotropy of the small scales of a turbulent flow is strongly violated when applied to passive scalars, with both experiments and direct numerical simulations (DNS) finding that the skewness of the scalar derivative remains of the order of unity when a large-scale mean scalar gradient is imposed, while it would be zero for a locally isotropic flow [3]}, [4]}, [5]}, [6]}. This strong violation of small-scale isotropy of the scalar field is often attributed to the presence of ramp-cliff structures in the scalar field through which large and small scales of the scalar field are directly connected [7]}, [8]}. It has also been found that intermittency in the scalar field is even stronger than that for the velocity field [9]}. Indeed, even for a stochastic model where the velocity field has Gaussian statistics, the scalar field has non-Gaussian statistics [10]}, [11]}, [12]}. Another profound difference is that while the velocity field exhibits a dissipation anomaly (i.e. the averaged turbulent kinetic energy dissipation rate is independent of viscosity for high Reynolds numbers), the scalar field does not, with the scalar dissipation rate decreasing as \(\sim 1/\log (Sc)\)  as the Schmidt number \(Sc\)  is increased [13]}. Interestingly, however, recent DNS results have shown that while the expected correspondence between the analogous velocity and scalar statistics is not observed for \(Sc\le 1\) , it is recovered for sufficiently large \(Sc\)  (even for \(Sc=7\) ) where a viscous-convective sub-range emerges in the scalar field [14]}.
